SPH for microfluidic suspensions: Surface tension, wetting and solid

Transcrição

SPH FOR MICROFLUIDIC SUSPENSIONS:
SURFACE TENSION, WETTING AND SOLID
PARTICLES
T. Breinlinger
30.06.2014
© Fraunhofer-Institut für Werkstoffmechanik IWM
SPH FOR MICROFLUIDIC SUSPENSIONS:
SURFACE TENSION, WETTING AND SOLID
PARTICLES
T. Breinlinger
30.06.2014
Microfluidic suspensions:
• Drops
• Particles
• Wetting/dewetting
• Drying
„ANTZ“ Universal Pictures
2
Group
Powders and fluidic systems
 Simulation of powdertechnological
processes and process chains
 Granulation
 Powder transfer
 Powder compaction
 Tape casting, screen printing
 Drying, debinding
 Sintering
 Simulation of manufacturing processes
 Modeling of complex suspensions
Dr. Torsten Kraft
Granulation via
spray drying
Abrasive processing
with bonded grain
Filling, compaction and
sintering of a ceramic
seal disc
 Microfluidic system design
 Determination of material data
 Analysis of various materials
Process optimization by simulation
3
Wetting of a
structured surface
Sintering warpage
of a printed LTCC
AGENDA
 Smoothed Particle Hydrodynamics (weakly compressible)
 Surface tension in SPH
 Different approaches
 Wetting in SPH
 How to incorporate wetting in surface tension models
 Applications
 Modeling suspensions with SPH
 Different approaches
 Applications
4
Weakly compressible SPH
Basic equations
J.J. Monaghan, Rep. Prog. Phys. 68 (2005)
A. Colagrossi et al., J. Comput. Phys. 191 (2003)
P.W. Cleary, Appl. Math. Model. 22 (1998)
S. Adami et al., J. Comput. Phys. 229 (2010)
 Density summation
𝜌𝑖 = 𝑚𝑖
𝑊𝑖𝑗
𝑗
 Momentum equation
𝜕𝒗𝑖
1
=
−𝛁𝑝𝒊 + 𝜂𝛁 2 𝒗𝑖 + 𝑭
𝜕𝑡
𝜌𝑖
 Pressure term
𝛁𝑝𝑖 =
 Viscous term
𝜂𝑖 𝛁 2 𝒗𝑖 =
 Surface tension force
𝑭𝑖
5
𝑠
=…
𝑗
𝑠
𝑖
+𝒈
𝑚𝑗
𝑝𝑖 + 𝑝𝑗
𝛁𝑊𝑗𝑖
𝜌𝑗
𝑗
𝜉
4𝜂𝑖 𝜂𝑗 𝒗𝑖 − 𝒗𝑗 ⋅ 𝒙𝑖 − 𝒙𝑗 𝑚𝑗
𝛁𝑊𝑖𝑗
𝜂𝑖 +𝜂𝑗 𝒙 − 𝒙 2 + 𝛽ℎ2 𝜌𝑗
𝑖
𝑗
Surface Tension in SPH
6
Surface tension
Introduction
 Caused by:
 Cohesive forces among liquid molecules.
 Effects:
 Contraction of liquid surface.
 Pressure jump across surface.
Δp = −𝜎 div 𝒏
𝜎: surface tension
𝒏: surface normal
7
Surface tension
Modeling techniques in SPH
Bas ed on caus e
 Pairwise forces
Bas ed of effect
 CSF model
(Continuum surface force)
S. Nugent and H.A. Posch, Phys. Rev. E 62 (2000)
J.U. Brackbill et al., J. Comput. Phys. 100 (1992)
A. Tartakovsky and P. Meakin, Phys. Rev. E 72 (2005)
J.P. Morris, Int. J. Numer. Meth. Fl. 33 (2000)
8
Surface tension
Implementation of pairwise forces in SPH
 Nugent & Posch:
𝑝=
𝜌𝑘𝑇
− 𝑎𝜌2
1 − 𝜌𝑏
van der Waals EOS
 Large neighborhood required.
 Tartakovsky & Meakin:
𝑭𝑖
𝑠
1.5𝜋
= 𝑠𝑖𝑗 𝑐𝑜𝑠
𝑟 − 𝑟𝑖
3ℎ 𝑗
𝑟𝑖𝑗
𝑟𝑖𝑗 ≤ ℎ
 Attractive at long range, repulsive at short range.
 Both models require case calibration.
 Just as high EOS stiffness, these forces can add molecular viscosity.
9
Surface tension
Implementation of CSF in SPH
𝑠
 Surface tension force
𝑭𝑖
 Color function
𝑐𝑖𝑗 =
 Gradient of
color function
𝛁𝑐𝑖 = 𝒏𝛿 =
 Surface normal
𝒏 = 𝛁𝑐 𝛁𝑐
 Surface curvature
10
= 𝜎𝜅𝒏𝛿
1, 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑝ℎ𝑎𝑠𝑒𝑠
0, 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑝ℎ𝑎𝑠𝑒.
𝜅 = 𝛁𝒏𝑖 = 𝑑
𝑗𝑑𝑖𝑓𝑓. 𝑝ℎ𝑎𝑠𝑒
𝑉𝑖 2 + 𝑉𝑗 2
𝑗
𝒏𝑖 − 𝒏𝑗 𝑉𝑗 𝛁𝑊𝑖𝑗
𝑗
𝒙𝑖 − 𝒙𝑗 𝑉𝑗 𝛁𝑊𝑖𝑗
𝜌𝑖
𝛁𝑊𝑖𝑗
𝜌𝑖 + 𝜌𝑗
Surface tension
Modeling techniques in SPH - Summary
Bas ed of effect
Bas ed on caus e
 Pairwise forces
 CSF model
+ Free surface possible
+ Contact angle and surface
tension as input parameter
+ Intrinsic wetting
-
Calibration required
-
„Raspberry“ clustering
-
Artificial viscosity
-
More complex to implement
-
Two phases required
J.P. Morris, Int. J. Numer. Meth. Fl. 33 (2000)
11
Wetting in SPH
(for CSF-based models)
12
Wetting
Contact line treatment
 2 phase problem governed by Young-Laplace equation
Δp = −𝜎 div 𝒏
Δp
 3 phase contact line governed by Young equation
𝜎 cos 𝜃𝑒𝑞 + 𝛾𝑠𝑙 − 𝛾𝑠𝑣 = 0
13
Wetting
 Normal correction method
 Prescribed surface normal near walls
𝒏𝑡𝑙 = 𝒏𝑡 sin 𝜃𝑒𝑞 −𝒏𝑤 cos 𝜃𝑒𝑞
No treatment
Prescribed normal
𝜃𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 𝜃𝑒𝑞 = 60°
14
Wetting
 Normal correction method
 Prescribed surface normal near walls
 What happens without additional treatment:
 Spurious currents
 Especially at triple line
 Even induced there?
15
Wetting
T. Breinlinger et al., J. Comput. Phys. 243 (2013)
 The normal correction should be smoothed
𝒏∗ 𝑖 =
𝑓𝑖 𝒏+ 1−𝑓𝑖 𝒏𝑡𝑙
𝑓𝑖 𝒏+ 1−𝑓𝑖 𝒏𝑡𝑙 𝑖
Sharp correction
with
𝑓𝑖 (𝑑𝑤,𝑖 ) = 𝑑𝑤,𝑖 𝑑𝑚𝑎𝑥
𝑑𝑤,𝑖 = 𝑚𝑖𝑛 𝒙𝑖𝑗 ⋅ 𝒏𝑤,𝑖 − 𝛿
Smoothed correction
𝜃𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 60°, 𝜃𝑒𝑞 = 90°
16
Wetting
Spurious currents
 The smoothed normal correction scheme helps to reduce spurious
currents
17
Wetting
Solution of Young-Laplace equation
𝜌𝑙 𝜌𝑔 = 1
𝜂𝑙 𝜂𝑔 = 1
𝑔=0
 3D drop on solid substrate (color code: pressure)
𝜃=60°
18
𝜃=150°
Wetting
Equilibrium shape of drops
 2D drop on solid substrate
𝜌𝑙 𝜌𝑔 = 1
𝑔=0
Sharp normal correction
19
Smoothed normal correction
Wetting
Pinning effects
 2D drop crossing an edge
𝜌𝑙 𝜌𝑔 = 1000
𝑔 = 0.007 … 0.014
20
Suspensions in SPH
21
Suspensions
Different approaches
 A suspension is a complex fluid consisting of a liquid phase and particles.
 Suitable approach for modeling depends on
 Level of detail
 System size
Homogeneous
Macroscopic
Mesoscopic
Microscopic
Level of detail
System size
22
Suspensions
Different approaches
Homogeneous approach:
 Describes suspension as homogeneous fluid with complex
rheological model.
Macroscopic approach:
 Uses two (or more) intersecting and interacting
continuous fluids to describe both phases.
Mesoscopic approach:
 Uses discrete element (DEM) model of solid particles and
local averaging for coupling with SPH.
Microscopic approach:
 Uses discrete solid particles and fully resolves the flow
around them (in CFD known as „immersed boundaries“).
23
A. Wonisch et al., J. Am. Ceram. Soc. 94 (2011)
Suspensions
Homogeneous approach
 Uses „regular“ SPH.
 Complex rheology:
 Shear thickening/thinning
Shear load
Shear thinning
Relaxing
Viscosity 𝜂
Viscosity 𝜂
 Thixotropy
Shear thickening
Shear rate 𝛾
Time 𝑡
 Can be implemented relatively easily in SPH as 𝜂 = 𝑓(𝛾) or
𝜂 = 𝑓 𝛾 in case of thixotropy.
24
Suspensions
Homogeneous approach
Example: Tape casting
Macroscopic shear rate
 SPH continuum simulation of slurry
-> shearrate 𝛾 in system is known.
 Jeffery equation
Φ 𝑡 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑟𝑒 tan
𝛾𝑡
𝑟𝑒 + 𝑟𝑒 −1
gives particle orientation for nonspherical particles.
 Including the shear history, this gives very
good argeement with the microstructure
found in experiments.
25
Microscopic orientation
Suspensions
Mesoscopic approach
 Discrete element method DEM solves Newton‘s equations for individual
solid particles.
 SPH solves Navier-Stokes equations for fluid phase.
 Coupled by local averaging.
 calculate coupling forces on DEM
vi
wj Rj
rj v j
ri wi
 integrate them on SPH
 normalize by 1 = 𝑚 𝜌 𝑊 based on DEM particles to ensure
conservation of momentum.
 The size ratio of DEM vs. SPH particles should not be too far from 1.
 DEM too small -> large summations -> slow.
 DEM too large -> averaging no longer justified.
26
T. Breinlinger et al., Proc. 6th SPHERIC (2011)
M. Robinson et al., Proc. 6th SPHERIC (2011)
D. Gao & A. Herbst, Int. J. Comp. Fluid Dyn. 23 (2009)
Suspensions
Mesoscopic approach
 Coupling SPH and DEM
 Coding perspective
SPH
Main loop
Calculate
density, stress
tensors etc.
Determine
Neighbors
DEM
Determine
volume
fractions
Determine
Neighbors
27
Calculate
interaction
forces
Integration
Calculate
coupling forces
Calculate
Overlaps
Calculate
interaction
forces
Integration
Suspensions
Mesoscopic approach
Example 2: Spray drying process - Drying of individual droplets
 Solid primary particles with DEM
 Cohesion depending on moisture
 Fluid solvent with SPH
 Newtonian fluid
 Interaction of DEM/SPH via coupling forces
 Drag, capillary
 Drying model in SPH
 Isothermal
 Discrete phase transition
Solvent
(SPH)
Droplet
28
Solid particles
(DEM)
Suspensions
Mesoscopic approach
Example 2: Spray drying process - Drying of individual droplets
 Results
Pure drop
Suspension drop
(SPH)
(SPH+DEM)
 Binary phase change works for pure SPH but induces buckling and
bulging in coupled DEM-SPH simulations.
29
Suspensions
Mesoscopic approach
Mesh based CFD as an alternative for spray drying process.
 Binary drying model in SPH causes instabilities.
 Mesh based Volume of Fluid (VOF) Method allows for continuous phase
change.
 VOF is implemented in OpenFOAM („interFOAM“).
 Implemented a customised coupled VOF-DEM solver in OpenFOAM.
 Momentum conservation
 DEM substepping in time
 Coupling forces for
capillary force
 Cohesion depending
on moisture
30
Suspensions
Mesoscopic approach
Mesh based CFD as an alternative for spray drying process.
 Results
 Drying and granulation can be simulated using VOF+DEM.
 Granule morphology depends on the relation of
cohesive vs. capillary forces.
cohesive vs. capillary forces:
strong
31
medium
weak
Suspensions
Microscopic approach
 Direct numerical simulation of suspensions with SPH
 Fluid via “regular” SPH:
appropriate properties
(density, viscosity etc.)
 Solid particles via rigid bodies
(rigid clusters of SPH particles)
 Modes of interaction
 Fluid-Fluid -> regular SPH
 Fluid-Solid -> regular SPH
 Solid-Solid -> hard spheres
32
Solid particles
Solvent
Suspensions
About the rigid clusters
 A rigid body k consists of a cluster of rigidly linked SPH-particles 𝑖 ∈ 𝑠𝑘
(sub-particles)
 Total cluster force is the sum of all sub-particles forces:
force on rigid body:
𝑓𝑘 =
𝑓𝑖
torque on rigid body: 𝑡𝑘
𝑖∈𝑆𝑘
=
𝑏𝑖 × 𝑓𝑖
𝑖∈𝑆𝑘
bi: position of the sub-particles relative to the
cluster center of mass
 Quaternion 𝑞𝑘 = 𝜉𝑘 , 𝜂𝑘 , 𝜁𝑘 , 𝜒𝑘 gives the cluster orientation
 Time integration of 𝑞𝑘 through rigid body solver:
1
𝑞𝑘 𝑡 + Δ𝑡 = 𝑞𝑘 𝑡 + 𝑞𝑘 𝑡 Δ𝑡 + 𝑞𝑘 (𝑡)Δ𝑡 2 − 𝜆𝑘 (𝑡)𝑞𝑘 (𝑡)∆𝑡 2
2
with 𝜆𝑘 (𝑡) being the Lagrangian multiplier satisfying 𝑞𝑘
33
2
𝑡 + Δ𝑡 = 1
I.P. Omelyan, Phys. Rev. E 58 (1998)
Suspensions
Example: platelet orientation
 2d parallelization using 12 CPUs
 3d representative volume cell with
Lees-Edwards-boundary-conditions
 Edge length: 250 µm
 Initial SPH-particle spacing Dx=5 µm
->125000 SPH particles
 h/Dx=1.05, qubic spline kernel
x
 Random cluster orientation
 Shear rate: 100 s-1
 Shear duration: 1 s
 Fluid viscosity: 0.42 Pa s
34
z
Suspensions
x
z
35
Conclusions
Surface tension & wetting
Suspensions
 Surface tension can be accurately
modeled using SPH.
 Different approaches available.
 Pairwise forces
 Complex rheology models can be
very useful for large scale
simulations.
 Intrinsic wetting
 Molecular viscosity
 Free surfaces
 CSF
 Technical parameters
 Wetting must be modelled
 Smoothed normal correction
should be used
36
 Level of detail vs. system size.
 Drying of a particle suspension
with SPH can be difficult.
 On the microscale, cluster-based
SPH modeling can be very
powerful.

SPH for microfluidic suspensions: Surface tension, wetting and solid

Transcrição

Documentos relacionados

estimation of hourly global radiation at campina grande

graciliano ramos vidas secas livro completo pdf pdf